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Liar's Poker

Rules

Liar's Poker is played using randomly picked currency
from your wallet. The denomination does not matter.
Hoarding ringers is strictly not allowed.

All players must agree on the stakes, for example $1
per person per round. You do not have to use the exact
bill you are playing for, for example you can use a $20
bill although playing for only $1.

A rule should be set for who goes first, for example
whose letter in the serial number is lowest, or who won
the last time. Who goes first is not very important, in
my opinion.

A hierarchy of numerals should be established. I
prefer zeros are low and nines are high.

Players in turn bid on the combined numbers in all
serial numbers, your own and those of the other
players.

Each player must in turn either declare a higher hand
than the player player or challenge.

In a 3+ player game all players must challenge to end
the game.

Eventually a player will be challenged. Then the
combined serial numbers will be used to determine if the
last hand called exists. For example if the challenged
hand is four eights then there must be at least four
eights on all serial numbers. If players trust each other
than can simply declare how many of the given number they
have, of course the challenged player reserves the right
to see the bills if he so requests.

If the serial numbers support the challenged player
then the player will win the agreed upon stakes from each
other player. Otherwise the challenged player must pay
each other player the agreed upon stakes.

Let's look an example. Suppose there are three players
playing for $1 stakes with the following serial numbers:

At this point there must be 6 zeros for player 3 to win.
There are only 5 so player 3 must play player 1 and 2 $1
each. Had player 2 had a zero then player 3 would have
won.

Strategy

In 3+ player games it often happens that a player is in a
damned if you do damned if you don't situation. Assuming
that by challenging you will definitely lose, and by raising
you definitely will be challenged, you should always raise
in a 2-player game, raise if your probably of winning by
doing is 25% or greater in a 3-player game, 33.33% in a
4-player game, and (n-2)/(2n-2) for n players. Of course
nothing is ever certain, so this scenario is admitedly
unrealistic.

It often happens that you need at least one other player
to have at least one of a certain number for you to win.
Assuming nothing about the other player's numbers (again an
admitedly unrealistic assumption) the following table shows
the probability of the total number of any given number
according to the number of other players.

Probabilities in Liars Poker

Number ofNumerals

Number of Other Players

1

2

3

4

0

0.430467

0.185302

0.079766

0.034337

1

0.382638

0.329426

0.212711

0.122087

2

0.148803

0.274522

0.271797

0.21026

3

0.033067

0.142344

0.221464

0.233622

4

0.004593

0.051402

0.129187

0.188196

So if you are playing with two other players and you have
3 fives and call four fives the probability of winning if
you are challenged is 1-0.185302 = 0.814698. However if you
need two fives the probability drops to 1-0.185302-0.329426
= 0.485272.

The next table shows the probability that any specific number will appear n times.

Specific Number Odds
in Liar's Poker

Number

Probability

8

0.00000001

7

0.00000072

6

0.00002268

5

0.00040824

4

0.00459270

3

0.03306744

2

0.14880348

1

0.38263752

0

0.43046721

Total

1.00000000

The next table shows the probability of every possible type of bill, categorized by the number of each n-of-a-kind. For example, the serial number 66847680 would have one three of a kind, one pair, and three singletons, for a probability of 0.1693440.

General Probabilities in Liar's Poker

8 o.a.k.

7 o.a.k.

6 o.a.k.

5 o.a.k.

4 o.a.k.

3 o.a.k.

2 o.a.k.

1 o.a.k.

Probability

1

0.0000001

1

1

0.0000072

1

1

0.0000252

1

2

0.0002016

1

1

0.0000504

1

1

1

0.0012096

1

3

0.0028224

2

0.0000315

1

1

1

0.0020160

1

2

0.0015120

1

1

2

0.0211680

1

4

0.0211680

2

1

0.0020160

2

2

0.0141120

1

2

1

0.0423360

1

1

3

0.1693440

1

5

0.0846720

4

0.0052920

3

2

0.1270080

2

4

0.3175200

1

6

0.1693440

8

0.0181440

Total

1.0000000

o.a.k. = "of a kind"

The next table summarizes the table above in groups of the more frequent occurrence of any digit.